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MOTION OF A PARTICLE 293
Substituting the last expression for h in that for P we obtain
V7 (M + m)
It will be noted that the period of revolution depends upon the major axis but not upon the minor axis of the orbit.
The results obtained in discussing the motion of two gravitating particles are as they appear to an observer who is located on one of the bodies. The form and size of the orbit, the period of revolution, etc., will be the same whether the observer is located on one or on the other of the two bodies. For instance, to an observer on the moon the earth describes an orbit which is exactly similar to the orbit which the moon appears to describe to an observer on the earth.
224. Mass of a Planet which has a Satellite. — In order to fix our ideas let the earth be the planet. Then, since the acceleration due to the sun is practically the same on the moon as it is on the earth, the period of revolution of the moon around the earth is the same as if they were not in the gravitational field of the sun. Therefore the period of the moon around the earth is
while that of the earth around the sun is p =
where M, m, and m! are the masses of the sun, of the earth, and of the moon, respectively, a is the semi-major axis of the earth's orbit, and o! that of the moon's orbit. Squaring these equations and dividing one by the other
m+m' _ /P\2 /flA3
M + w \P7 \a) '